We conducted the experiments on the FaceScape dataset [1]}, a large-scale 3D human face dataset consisting of \(16,940\)  topologically uniformly registered 3D face meshes along with high-quality textures. The dataset contains 847 identities with 20 expressions per identity, and each mesh consists of \(26,317\)  vertices of which 68 are 3D landmark vertices covering eyes, nose, mouth, jaw, and eyebrows. With some abuse of notation, we denote a 3D face mesh as \(\mathbf {x}\in \mathbb {R}^{3V}\)  (recall that \(\mathbf {x}\)  is used to represent an image in other parts of this paper), where \(V = 26,317\)  is the number of vertices. Given an index \(l\)  of a vertex (e.g., one of the 68 landmark vertices), the 3D coordinate of this vertex is defined as \(\mathbf {x}[l] \in \mathbb {R}^{3}\) . We pre-train a generative mesh model \(G\)  that maps each latent code \(\mathbf {z}\sim \mathcal {N}(\mathbf {0}, \mathbf {I})\)  to a 3D mesh \(\mathbf {x}\) .
PromptGen allows us to obtain geometric control on the 3D meshes.
Specifically, given two landmark vertex indices \(l_1\)  and \(l_2\) , we define an intra-sample constraint that enforces the signed distance (along a given direction) between the two vertices to be \(s\) .
To this effect, we define the signed-distance energy as
\(E_{\text{signed-distance}}(\mathbf {x}) = |d(\mathbf {v}_1, \mathbf {v}_2) - s|, \quad \mathbf {v}_1 = \mathbf {x}[l_1], \mathbf {v}_2 = \mathbf {x}[l_2]\) 
